Lesson 6: Rationalization

When we rationalize a denominator, we are trying to find an equivalent expression without a square root in the denominator. To rationalize the denominator, first identify the form from the chart below, then multiply by the given rational expression. $A$, $B$ and $C$ represent any non-zero expressions.

Form	Multiply by
$\displaystyle \frac{A}{\sqrt{B}}$	$\displaystyle \frac{\sqrt{B}}{\sqrt{B}}$
$\displaystyle \frac{A}{\sqrt{B} \pm \sqrt{C}}$	$\displaystyle \frac{\sqrt{B} \textrm{ }\mp \textrm{ }\sqrt{C}}{\sqrt{B} \textrm{ }\mp \textrm{ }\sqrt{C}}$
$\displaystyle \frac{A}{\sqrt{B} \pm C}$	$\displaystyle \frac{\sqrt{B}\textrm{ }\mp \textrm{ }C}{\sqrt{B}\textrm{ }\mp \textrm{ }C}$

When we rationalize a numerator, we are trying to find an equivalent expression without a square root in the numerator. To rationalize the numerator, first identify the form from the chart below, then multiply by the given rational expression. $A$, $B$ and $C$ represent any non-zero expressions.

Form	Multiply by
$\displaystyle \frac{\sqrt{A}}{B}$	$\displaystyle \frac{\sqrt{A}}{\sqrt{A}}$
$\displaystyle \frac{\sqrt{B}\pm\sqrt{C}}{A}$	$\displaystyle \frac{\sqrt{B} \textrm{ }\mp \textrm{ }\sqrt{C}}{\sqrt{B} \textrm{ }\mp \textrm{ }\sqrt{C}}$
$\displaystyle \frac{\sqrt{B}\pm C}{A}$	$\displaystyle \frac{\sqrt{B}\textrm{ }\mp \textrm{ }C}{\sqrt{B}\textrm{ }\mp \textrm{ }C}$